Centre of mass decoherence due to time dilation: paradoxical - - PowerPoint PPT Presentation

Centre of mass decoherence due to time dilation: paradoxical frame-dependence

Lajos Diósi Wigner Research Centre for Physics H-1525 Budapest 114, POB 49, Hungary 16 Sept 2016, Castiglioncello Acknowledgements go to: EU COST Action MP1209 ‘Thermodynamics in the quantum regime’

Two stories for one model Newtonian Equivalence Principle Relativistically: c.o.m. couples to internal d.o.f. C.o.m. positional decoherence due to g Frame-dependence of positional decoherence? Frame-dependence of positional decoherence! Summary: Pikovski et al. theory for pedestrians

Two stories for one model

Effect: Positional decoherence of composite objects, ∝ g/c2. Pikovski-Zych-Costa-Brukner, Nature Phys. 11, 668 (2015).

◮ Method: 1/c2 GR correction to composite object QM. ◮ Arguments: relativistic, semiclassical ◮ Claim: universal decoherence due to gravitational time

dilation Same Hamiltonian, pedestrian story [L.D. arXiv:1507.05828]:

◮ Method: 1/c2 SR correction to composite object QM. ◮ Arguments: non-relativistic, exact dynamics ◮ Claim: frame-dependent decoherence due to 1/c2

coupling between c.o.m. and i.d.o.f. SR/GR arguments for frame-dependence: Bonder-Okun-Sudarski PRD92, 124050, (2015) Pang-Chen-Khalili PRL117, 090401 (2016)

Newtonian Equivalence Principle

http://wigner.mta.hu/∼diosi/tutorial/freefalltutor.pdf Free-Falling observer: g = 0. Laboratory observer: g = 9.81cm/s2. Example: center-of-mass (c.o.m.) motion of free mass m. Free-Falling: x, p;

• H0 =
• p2

2m Laboratory: X, P;

• Hg =
• P2

2m + mg X (X : vertical) Canonical transformation:

• U = exp
• −

igt2 p/2

• exp
• imgt

x

• exp
• img2t3/6
• X =

U x U† = x − gt2/2

• P =

U p U† = p − mgt

• Hg =

U H0 U† − i ˙

• U

U†

Relativistically: c.o.m. couples to internal d.o.f.

Internal Hamiltonian Hi is additive: Htot

0/g =

H0/g + Hi. Special relativistic correction, try m → m + Hi/c2. Free-Falling: x, p,

• i;
• Htot

=

• p2

2(m + Hi/c2) + Hi Laboratory: X, P, Oi;

• Htot

g

=

• P2

2(m + Hi/c2) +(m+ Hi/c2)g X+ Hi Canonical transformation U (as before, just m→m+ Hi/ c2):

• X =

U x U† = x −gt2/2 pure kinematics, as before

• P =

U P U† = p − (m+ Hi/c2)gt mixing i.d.o.f. to p

• Oi =

U

• i

U† =exp(ic−2gt Hi x)

• i exp(

− ic−2gt Hi x) mixing x to i.d.o.f. Note: U Hi U† = Hi.

C.o.m. positional decoherence due to g

• Htot

g

=

• P2

2m + g c2

• X

Hi + Hi A wonderful coupling betwen Laboratory c.o.m. X and Hi. If initial state ρtot = ρcm ⊗ ρi where ρi = Z −1 exp(−β Hi), that’s typical system-bath situation, yields c.o.m. positional decoherence: x1| ρcm(t) |x2 ≈ e− 1

2 t2/τ 2 dec × x1− 1 2gt2|

ρcm(0)|x2− 1

2gt2

decoherence rate: 1 τdec = g c2

• kBCT|x1 − x2|.

m=1µg, C=10−5cal/K, T=300K, x1−x2 =1µm: ⇒ τdec ∼1ms.

◮ Positional decoherence ∝g in Laboratory frame ◮ No positional decoherence in Free-Fall frame

Frame-dependence of positional decoherence?

Hm ..., that’s counterintuitive. If |x1 + |x2 decays in the Laboratory and |X = |x − 1

2gt2

then in the Free-Fall frame |X1 + |X2 should, too, decay. This argument is just false: |X = |x − 1

2gt2.

No closed map exists between Laboratory eigenstates |x and Free-Fall eigenstates |X! Why:

• X =

U x U† = x −gt2/2 pure kinematics

• P =

U P U† = p − (m+ Hi/c2)gt mixing i.d.o.f. to p C.o.m. generic observables are frame-dependent. Split Hcm ⊗ Hi is frame-dependent. Hilbert space Hcm is frame-dependent. You don’t expect this. It is just so if you start with

• Htot

FF =

• p2

2(m + Hi/c2) + Hi and change for Laboratory frame, or vice versa.

Frame-dependence of positional decoherence!

Yes! In Earth gravity g:

◮ Free-Falling screen detects no decoherence ◮ Laboratory (fixed) screen detects positional decoherence

In gravity-free (g = 0) frame:

◮ Static screen detects no decoherence ◮ Accelerated screen detects positional decoherence

Lucid proof: Pang-Chen-Khalili [PRL 117, 090401 (2016)]: x x v x L p

2 1

screen

Fringes shifted ∝ arrival time: cos

p(x1 − x2)/L

• xscreen − vscreen

Lm p

• m is random since m→m+Hi/

c2. Visibility supressed ∝ vscreen. Choice vscreen=gt recovers τdec just like in Earth’s Laboratory frame.

Summary: Pikovski et al. theory for pedestrians

Pedestrian=non-relativistic thinker, sees different depths. i) SR (not GR) correction to standard Hamiltonian:

• H =
• p2

2(m + Hi/c2) + Hi A piece of SR, but no Lorentz inv., no general cov. ii) Exact Galilean inv. and Newtonian Equivalence Principle. iii) We can interpret everything in non-relativistic terms - plus the fact that m contains the correction Hi/c2. iv) Positional decoherence is missing in inertial frames. It emerges in accelerating frames only. v) Moving (v ≪ c) detector sees different interference fringes, accelerating detector sees same fringe as static one in gravity. With these pedestrian lessons can we put the theory back to SR/GR context (and re-attribute positional decoherence to time dilation).